Nuprl Lemma : csm-ap-cubical-app
∀[X,Delta:CubicalSet]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[w:{X ⊢ _:ΠA B}]. ∀[u:{X ⊢ _:A}]. ∀[s:Delta ⟶ X].
  ((app(w; u))s = app((w)s; (u)s) ∈ {Delta ⊢ _:((B)[u])s})
Proof
Definitions occuring in Statement : 
cubical-app: app(w; u)
, 
cubical-pi: ΠA B
, 
csm-id-adjoin: [u]
, 
cube-context-adjoin: X.A
, 
csm-ap-term: (t)s
, 
cubical-term: {X ⊢ _:AF}
, 
csm-ap-type: (AF)s
, 
cubical-type: {X ⊢ _}
, 
cube-set-map: A ⟶ B
, 
cubical-set: CubicalSet
, 
uall: ∀[x:A]. B[x]
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
uimplies: b supposing a
, 
cubical-type: {X ⊢ _}
, 
cubical-term: {X ⊢ _:AF}
, 
cube-set-map: A ⟶ B
, 
nat-trans: nat-trans(C;D;F;G)
, 
csm-ap-term: (t)s
, 
cubical-app: app(w; u)
, 
csm-id-adjoin: [u]
, 
csm-ap-type: (AF)s
, 
pi1: fst(t)
, 
csm-ap: (s)x
, 
csm-adjoin: (s;u)
, 
cubical-set: CubicalSet
, 
functor-arrow: arrow(F)
, 
functor-ob: ob(F)
, 
type-cat: TypeCat
, 
cat-comp: cat-comp(C)
, 
cat-arrow: cat-arrow(C)
, 
name-cat: NameCat
, 
cat-ob: cat-ob(C)
, 
pi2: snd(t)
, 
I-cube: X(I)
, 
cubical-pi: ΠA B
, 
csm-id: 1(X)
, 
all: ∀x:A. B[x]
, 
top: Top
, 
cat-id: cat-id(C)
, 
implies: P 
⇒ Q
, 
prop: ℙ
, 
subtype_rel: A ⊆r B
, 
and: P ∧ Q
, 
cubical-pi-family: cubical-pi-family(X;A;B;I;a)
, 
cubical-type-at: A(a)
, 
squash: ↓T
, 
label: ...$L... t
, 
true: True
, 
guard: {T}
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
cc-adjoin-cube: (v;u)
Lemmas referenced : 
cubical-term-equal, 
csm-ap-type_wf, 
cube-context-adjoin_wf, 
csm-id-adjoin_wf, 
csm-ap-term_wf, 
cubical-app_wf, 
cube-set-map_wf, 
cubical-term_wf, 
cubical-pi_wf, 
cubical-type_wf, 
cubical-set_wf, 
ob_pair_lemma, 
ident_trans_ap_lemma, 
list_wf, 
coordinate_name_wf, 
I-cube_wf, 
equal_wf, 
cubical-pi-family_wf, 
id-morph_wf, 
subtype_rel-equal, 
cube-set-restriction_wf, 
squash_wf, 
true_wf, 
cube-set-restriction-id, 
subtype_rel_self, 
iff_weakening_equal, 
cc-adjoin-cube_wf, 
cubical-type-at_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
hypothesis, 
independent_isectElimination, 
sqequalRule, 
isect_memberEquality, 
axiomEquality, 
because_Cache, 
setElimination, 
rename, 
productElimination, 
dependent_functionElimination, 
voidElimination, 
voidEquality, 
lambdaEquality, 
applyEquality, 
lambdaFormation, 
equalityTransitivity, 
equalitySymmetry, 
independent_functionElimination, 
functionExtensionality, 
instantiate, 
imageElimination, 
universeEquality, 
hyp_replacement, 
natural_numberEquality, 
imageMemberEquality, 
baseClosed
Latex:
\mforall{}[X,Delta:CubicalSet].  \mforall{}[A:\{X  \mvdash{}  \_\}].  \mforall{}[B:\{X.A  \mvdash{}  \_\}].  \mforall{}[w:\{X  \mvdash{}  \_:\mPi{}A  B\}].  \mforall{}[u:\{X  \mvdash{}  \_:A\}].
\mforall{}[s:Delta  {}\mrightarrow{}  X].
    ((app(w;  u))s  =  app((w)s;  (u)s))
Date html generated:
2018_05_23-PM-06_31_54
Last ObjectModification:
2018_05_20-PM-04_20_44
Theory : cubical!sets
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